Automatic real analyticity and a regal proof of a commutative multivariate Löwner theorem

We adapt the “royal road” method used to simplify automatic analyticity theorems in noncommutative function theory several complex variables. show that certain families of functions must be real analytic if they have nice properties on one-dimensional slices. Let E ⊂ R d E \subset \mathbb {R}^d open. A alttext="f colon right-arrow R"> f : → 1 t stretchy="false">) , …<!-- … encoding="application/x-tex">f(\varphi _1(t), \ldots , \varphi _d(t)) a matrix alttext="t"> encoding="application/x-tex">t whenever alttext="t element-of 0 ∈<!-- ∈ <mml:mn>0 encoding="application/x-tex">t \in (0,1) , alttext="phi i"> i encoding="application/x-tex">\varphi _i are automorphisms half plane, and tuple alttext="left-parenthesis encoding="application/x-tex">(\varphi maps encoding="application/x-tex">(0,1) into E"> encoding="application/x-tex">E . use road" lite only it analytically continues multivariate plane as map plane. Moreover, two variables locally sense Agler-McCarthy-Young.